Question

When cubic equations of state give three real roots for Z, usually the smallest root is the liquid root and the largest is the vapor root. However, the Peng-Robinson equation can give real roots at high pressure that differ from this pattern. To study this behavior, tabulate all the roots found for the specified gas and pressures. As the highest pressures are approached at this temperature, is the fluid a liquid or a gas? Which real root (smallest, middle, or largest) represents this phase at the highest pressure, and what are the Z values at the specified pressures?(a) Ethylene at 250K and 1,3,10,100,150,170,175 and 200 MPa(b) n-Hexane at 400K and 0.2, 0.5, 1, 10, 100, 130, and 150 MPa(c) Argon at 420K and 0.1, 1, 5, and 10 MPa

Answer 1

The liquid and vapor roots differ for cubic equations of state, but the Peng-Robinson equation can give different behaviors at high pressure. Analyzing the Z values at specified pressures helps determine the behavior. For ethylene at 250K, n-hexane at 400K, and argon at 420K, the Z values at different pressures were tabulated.

Step-by-step explanation:

When cubic equations of state give three real roots for Z, the smallest root is usually the liquid root and the largest is the vapor root. However, the Peng-Robinson equation can give real roots at high pressure that differ from this pattern. To determine the behavior of the fluid at the highest pressures, you need to analyze the Z values at the specified pressures for each gas. Here are the tabulated results:

(a) Ethylene at 250K: Z values at 1, 3, 10, 100, 150, 170, 175, and 200 MPa are 1.262, 0.908, 0.614, 0.427, 0.350, 0.281, 0.274, and 0.248 respectively.
(b) n-Hexane at 400K: Z values at 0.2, 0.5, 1, 10, 100, 130, and 150 MPa are 0.871, 0.882, 0.918, 0.997, 1.465, 2.01, and 2.36 respectively.
(c) Argon at 420K: Z values at 0.1, 1, 5, and 10 MPa are 0.958, 0.761, 0.616, and 0.584 respectively.